2
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Could you please try to find the intersections points of this two functions: $ y = x $ and $y=(a+x) \ {\rm e}^ {-2(a+x)} $ with $x\geq0$

Thanks for your help

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3
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To visualize the pairs of real numbers {x, a} that satisfy the equation,

you can use ContourPlot:

ContourPlot[(a + x) E^(-2 (a + x)) == x, {x, -2, 1}, {a, 0, 5}]

enter image description here

or RegionPlot with options MeshFunctions + Mesh:

RegionPlot[True, {x, -2, 1}, {a, 0, 5}, 
 BoundaryStyle -> None, 
 MeshFunctions -> {(#2 + #) E^(-2 (#2 + #)) - # &}, 
 Mesh -> {{0}}, 
 MeshStyle -> Directive[Thick, Red], 
 PlotStyle -> None, 
 PlotPoints -> 100]

enter image description here

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1
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Looks like you'll need some constraints on a.

a = -1;
f[x_] := (a + x) E^(-2 (a + x))
Plot[{x, f[x]}, {x, 0.001, 4}, PlotRange -> {Automatic, {-2, 1}}]

enter image description here

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0
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You can also use FindInstance. For example, for a given a:

FindInstance[{(a + x) Exp[-2 (a + x)] == x, a == 5}, {a, x}, Reals, 3]

enter image description here

Or for a given range for a:

FindInstance[{(a + x) Exp[-2 (a + x)] == x, 0 < a < 5}, {a, x}, Reals, 10]

enter image description here

Unfortunately, with Mathematica you can not get the solution, so I added the calculation with Maple here.

enter image description here

Test:

eq /. {a -> 1, x -> 0.12}
0.12
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